Logarithmic-dimensional convergence of mean-field BBVI

Let BBVI denote black-box variational inference, and let the mean-field variational family be the class of variational distributions whose coordinates are independent. Under mild assumptions, the dimensional-dependence conjecture. BBVI for the mean-field variational family converges with only logarithmic dimensional dependence or no explicit dimensional dependence at all. The paper has established an $\mathcal{O}\left(\sqrt{d}\right)$ dimension dependence for this family, while empirical results suggest that this can be improved; the precise mild assumptions and the stronger dimensional dependence remain open.